If $f:A\to \mathbb{R}$ is a continuous and one-to-one function, is $f$ strictly monotone?

Question

Theorem (1): Let $A \subseteq R$ be an interval and $f:A\to \mathbb{R}$ be a continuous one-to-one function. Then $f$ is strictly monotone.

Theorem (2): Let $A \subseteq \mathbb{R}$ be an arbitrary set and $f:A\to \mathbb{R}$ be a continuous and one-to-one function. Then $f$ is strictly monotone.

Is Theorem (2) right?

Answer 1

No. Suppose $A=[0,1)\cup(1,2]$ and define $f:A\to\mathbb R$ by $$f(x)= \begin{cases} x+1,& \textrm{if }x\in[0,1)\\ x-1,& \textrm{if }x\in(1,2] \end{cases} $$ The problem arises from the disconnectedness of $A$.